In analogy with the Onsager conjecture for Euler equations,
it was conjectured in [1]} that,
\(L^3_{t,x}\)  (resp. \(L^3_t B^0_{3,c_0,x}\) ) is the critical space for the conservation of the magnetic helicity.
More precisely,
any weak solutions belonging to \(L^3_{t,x}\)  (resp. \(L^3_t B^0_{3,c_0,x}\) ) conserve the magnetic helicity
(the rigidity part),
while below the threshold there exist weak solutions violating the magnetic helicity conservation
(the flexible part).
Moreover, the space \(C^{1/3}_{t,x}\)  (resp. \(L^3_t B^{1/3}_{3,{\infty },x}\) )
was conjectured to be the threshold for the
conservation of the total energy and the cross helicity
(see Remark REF  (ii) below).
